Mathematics: The Specialization of Mathematics

Scientific Thought: In Context
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Mathematics: The Specialization of Mathematics

Introduction

Mathematics is the study of relationships among and operations on abstract objects that obey definite rules, including numbers, variables, functions, rules, spaces, shapes, and sets. In its ancient origins, mathematics was concerned solely with numbers and geometry (the properties of definite shapes), which arose from measurable and countable phenomena and could, in part, be applied directly to business and architecture. During the nineteenth century, an increasing number of mathematicians became fascinated by relationships of pure reason and by the deductions that could be drawn from those relationships, even where these results seemed to have no application to the real world. This formalization of symbolic logic and abstract reasoning allowed mathematicians to develop the definitions, relations, and theorems of pure mathematics, but also—unexpectedly—had the effect of advancing applied mathematics, namely, those mathematical methods useful to science, engineering, and economics.

In both pure and applied mathematics, nineteenth century mathematicians took on increasingly specialized roles corresponding to the rapid compartmentalization and specialization of mathematics in general. One no longer simply became a mathematician, but a specialist in some particular branch of mathematics. As in other fields, specialization in mathematics has allowed the individual worker to gain a deeper knowledge in some particular area, but at the expense of breadth: a quip often attributed to German biologist Konrad Lorenz (1903–1989) states, “A specialist knows more and more about less and less and finally knows everything about nothing.” Overall, however, specialization in mathematics has been both unavoidable and beneficial.

Often, methods developed for the sake of specialized branches of pure mathematics have migrated to applied mathematics and become essential to new technologies and industries: for example, studies in abstract logic became, many decades after their original conception, the basis for modern digital computers.

Historical Background and Scientific Foundations

Starting in the late 1700s, the emerging European industrial revolution gave rise to new methods and knowledge in physics, astronomy, and engineering. These innovations created a spate of novel mathematical challenges. In the 1800s, mathematicians scrambled to invent and refine analytical methods to solve the seemingly endless list of questions and problems being raised by scientists and engineers. By the middle of the century, attention had begun to shift toward the operations of mathematical logic, considered on its own abstract terms. The result was an increased emphasis on the relationships and rules for evaluating axioms and postulates, which are the fundamental statements or claims on which structures of logical reasoning are raised.

Building on the calculus invented by innovators Sir Isaac Newton (1643–1727) of England and Gottfried Wilhelm von Leibniz (1646–1716) of Germany in the 1600s, nineteenth-century mathematicians extended the accuracy and precision of mathematical calculations. The increasingly complex and powerful structure of applied mathematics also depended on foundations laid by Swiss mathematician Leonhard Euler (1707–1783) in mechanics, differential and integral calculus, geometry, and algebra. Without all these tools, science and technology could not have continued to progress. The large strides in applied mathematics, however, left much theoretical and logical ground untouched. Mathematics relied on reasoning, but what was reason? How did one do it, and what were its rules, varieties, and limitations? After centuries of emphasis on practical applications, the nineteenth century was ripe for the development of a pure mathematics—a mathematics of pure reason.

With the nineteenth-century explosion in mathematical knowledge came, inevitably, specialization: the amount of total mathematical knowledge had become too large for any one person to master, the literature too large to read in a lifetime. Similar specialization was also occurring in the physical sciences. A result of the increasing specialization of mathematics was a schism between pure and applied mathematics. One definition of this division is that pure mathematics is advanced for theoretical reasons—it is concerned only with the mathematical validity of certain abstract, symbolic structures—while applied mathematics develops tools and techniques to solve problems in science, engineering, and economics. Such a simplistic definition, while containing much of the truth, omits the common history of and frequent crossovers between the two types of mathematics.

Starting in the nineteenth century, there evolved a profound difference in the methodology—the whole manner of proceeding—of pure and applied mathematics. In pure mathematics, deductions are valid if properly derived from given axioms and postulates. In the reasoning of applied mathematics, hypotheses (proposed explanations) are grounded in experimental evidence, cast in mathematical terms, and tested against further evidence. The choice of what mathematics to apply is based on observation, testing, and (often, in the earlier stages) speculation; some forms of mathematics will work, in practice, better than others, and one must find out by trial which these are. Pure mathematics is a matter of correctly following the laws of reasoning to produce a certain structure of mathematical relationships, whose interest is judged solely by its relationship to other mathematical structures. In applied mathematics, one seeks to discover which structure of mathematical relationships can be used to describe a recurrent pattern of natural phenomena (that is, a law of nature).

The professional split between the pure and applied math did not occur all at once. For example, the theories of German mathematician and physicist Johann Carl Friedrich Gauss (1777–1855) embraced and embodied both pure and applied mathematical concepts. Gauss' practical mathematical discoveries included advances in the study of the shape of the Earth (geodesy), planetary orbits, and statistical methodology (i.e., least-squares methods). Gauss also advanced pure mathematics through seminal work in number theory, representations of complex numbers, quadratic reciprocity, and a proof of the fundamental theorem of algebra.

In 1847, English mathematician George Boole (1815–1864) published his Mathematical Analysis of Logic. He followed it in 1854 with another important publication, The Laws of Thought. Boole asserted that the development of symbolic logic and pure mathematical reasoning was being retarded by an overdependence on applied mathematics. He advocated the expression of logical propositions in symbols used to represent logical statements: logical propositions could then be proved or disproved just like any other mathematical statements. Boole championed pure mathematical reasoning by attempting to dissociate abstract mathematical operations from any specific applications.

Boole's publication drew a line in the sand for mathematicians and highlighted a trend away from Gauss-like mathematical universalism and toward increased specialization within the profession of mathematics. In particular, there was an increasing divergence between pure and applied mathematics. As a consequence of the popularity of formalism, such as was advocated by Boole, there also resulted an increasing number of mathematicians dedicated to pure mathematics, with minimal consequences—or so it appeared—for science and technology.

This divergence was not always viewed with favor. Although initially created as a subdiscipline of mathematics, the field of mathematical logic was widely ignored or held in disdain by many mathematicians. By the end of the century, however, symbolic logic progressed from academic obscurity to popular entertainment. The books of English mathematician Lewis Carroll (1831–1898) on logic, The Game of Logic (1887) and Symbolic Logic (1896), became popular topics of conversation and sources of entertainment both for scholars and laymen in Victorian England; his most famous work, Alice in Wonderland (1865), contained much logic-play and has been quoted by scientists and philosophers for over a century. Despite initial resistance, mathematical logic made strong inroads into philosophy after William Stanley Jevons (1835–1882) heralded its use in his widely read Principles of Science (1874). Jevon argued

(correctly, as it turned out) that symbolic logic would be of importance to both philosophy and mathematics.

The increasing volume of work relating to number theory also lead to hierarchies with the emerging specialties of mathematics. Fueled by Gauss' work on the theory of numbers, algebraic theories of numbers took on a preeminent position in pure mathematics.

Some initially pure mathematical theory was met with outright derision and scorn. Among the most controversial of advances in mid-nineteenth century mathematics was the publication of non-Euclidean geometries by German mathematician Georg Friedrich Bernhard Riemann (1826–1866). Riemann asserted that Euclidian geometry—the ordinary geometry of triangles, lines, planes, and so forth, as taught in schools and used throughout architecture, surveying, and technology—was but one possible geometry, and that many others could be validly conceived. His expanded concepts of geometry treated the properties of curved space and seemed useless to nineteenth century Newtonian physicists. Many mathematicians also thought Riemann's conceptualizations bizarre.

Regardless, in the next century his theories proved of enormous consequence and value to the expansion of concepts of gravity and electromagnetism and of fundamental importance to the twentieth-century theoretical work of Albert Einstein (1879–1955) and others regarding the nature of space, time, and gravity. Topology as a specialization of mathematics (the study of the general properties of surfaces apart from changes in size or shape) was born in the advances of nineteenth century geometry and has also proved indispensable to modern physics.

Later in the nineteenth century, when Russian mathematician Georg Cantor (1845–1918) proposed his transfinite set theory, many thought it the height of abstraction. Advances in twentieth century physics, however, have also found use of Cantor's theories.

Yet pure and applied mathematics never became completely disconnected from each other: the challenges of real-world work have spurred developments in pure mathematics, while the abstract structures of pure mathematics have repeatedly been found necessary, sometimes many years after their development, for the description of the real world. Thus, not all developments in mathematics were polarized into the pure and applied camps. Early in the nineteenth century, the work of French mathematician Jean Baptiste Joseph Fourier (1768–1830) with mathematical analysis allowed him to establish what is known to modern mathematicians as the Fourier series, which is central to Fourier analysis, an important tool for both pure and applied mathematicians. And, late in the nineteenth century, group theory made possible the unification of many aspects of geometric and algebraic analysis.

Although there was an increasing trend toward specialization throughout the 1800s, near the end of the century French mathematician Jules Henri Poincaré (1854–1912) embodied Gauss's universalist spirit. Poincaré's work touched on almost all fields of mathematics. His insights provided significant advances in applied mathematics, physics, analysis, functions, differential equations, probability theory, topology, and the philosophical foundations of mathematics. Poincaré's studies of the chaotic behavior of systems subsequently provided the theoretical base for the continually evolving—and deeply practical—chaos theory of recent mathematics.

Mathematical rigor—the use of absolutely precise and unambiguous fundamental concepts and definitions—was required in the early part of the century by the extension and refinement of the calculus and was broadened near the end of the century by German mathematician Karl Theodor Wilhelm Weierstrass (1815–1897) to the types of analysis familiar to modern mathematicians (e.g., convergence series, theories using Abelian principles, periodic and elliptic functions, bilinear and quadratic forms).

The advancement of elliptic functions, principally through the work of Norwegian mathematician Neils Henrich Abel (1802–1829) and Prussian mathematician Karl Gustav Jacob Jacobi (1804–1851), provided mathematical precision in calculations required for discoveries in astronomy, physics, algebraic geometry and topology. In addition to their use in applied mathematics, however, the development of the theory of elliptic functions also spurred the study of functions of complex variables and provided a bridge between the widening chasm opening between pure and applied mathematics.

Modern Cultural Connections

Although there were subtle divisions of mathematics at the beginning of the nineteenth century, by the early twentieth century there were full and formal divisions of pure and applied mathematics and of subfields within each. University appointments and coursework syllabi began to reflect these divisions and an increasing number of professorial positions were designated for pure or applied mathematicians. Within each field, there was continuing sub-specialization as the amount of material to be mastered in each area grew over time. Yet the old arguments about the relative merits of pure and applied mathematics died out and remained, for the most part, dead: both types of mathematics offered ample challenge for the most brilliant minds.

One of the most dramatic examples of pure mathematics becoming essential in real life is the symbolic logic of Boole. Boole himself foresaw no practical uses for his methods, and his work accelerated the nineteenth-century trend toward a split between applied and pure mathematics—yet direct technological application of Boolean algebra in the digital computer has revolutionized warfare, personal life, finance, and science in the modern industrialized world. Boolean algebra is used by computer designers to specify the physical circuits that handle the billions of on-off signals that comprise the working information inside a digital computer. Also, as mentioned earlier, the non-Euclidean geometries of nineteenth-century pure mathematics have become everyday tools of modern physics.

Yet pure mathematicians do not pursue their craft for the sake of producing such tools: there is simply no way to tell which purely mathematical structures will someday find practical use and which will not. Instead, pure mathematicians pursue their theories with an interest that has often been compared to the creation of music or art. In doing so, they expand the range and rigor of human thought, expanding our intellectual possibilities: world-changing technological and scientific gains are a frequent side-effect of that drive.

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